where \(T_{i,n}\)  denotes the number of times arm \(i\)  was pulled in the first \(n\)  rounds. This is a natural definition based on regret decomposition [1]}, and in fact corresponds to the regret decomposition in the case \(\gamma \rightarrow 0^+\)  (i.e. the risk-neutral setting). In the following, we design and analyse ERTS, which aims to minimise \(R_n(\pi )\) , and also attain an instance-dependent lower bound, which establishes asymptotic optimality.
